On the zeros of partition functions with multi-spin interactions

Abstract

Let $X_1,\dots ,X_n$ be probability spaces, let $X$ be their direct product, let ${\varphi }_1,\dots ,{\varphi }_m:X\to \mathbb{C}$ be random variables, each depending only on a few coordinates of $x=(x_1,\dots ,x_n)$, and let $f={\varphi }_1+\cdots +{\varphi }_m$. The expectation $\mathbf{E}{e}^{\lambda f}$, where $\lambda \in \mathbb{C}$, appears in statistical physics as the partition function of a system with multi-spin interactions, and also in combinatorics and computer science, where it is known as the partition function of edge-coloring models, tensor network contractions, or a Holant polynomial. Assuming that each ${\varphi }_i$ is 1-Lipschitz in the Hamming metric of $X$, that each ${\varphi }_i(x)$ depends on at most $r\ge 2$ coordinates $x_1,\dots ,x_n$ of $x\in X$, and that for each $j$ there are at most $c\ge 1$ functions ${\varphi }_i$ that depend on the coordinate $x_j$, we prove that $\mathbf{E}{e}^{\lambda f}\ne 0$ provided $|\lambda |\le (3c\sqrt{r-1}{)}^{-1}$ and that the bound is sharp up to a constant factor. Taking a scaling limit, we prove a similar result for functions ${\varphi }_1,\dots ,{\varphi }_m:{\mathbb{R}}^n\to \mathbb{C}$ that are 1-Lipschitz in the ${\ell }^1$ metric of ${\mathbb{R}}^n$ and where the expectation is taken with respect to the standard Gaussian measure in ${\mathbb{R}}^n$. As a corollary, the value of the expectation can be efficiently approximated, provided $\lambda$ lies in a slightly smaller disc.